Examples of the successful stabilizing of period-1 and period-2 orbits are shown in Fig. 2.
Figure: The minima in anodic current plotted over a time
period during
which the control algorithm is switched on and off twice. The perturbations
added to the anodic potential to maintain control are shown in the bottom
graph. (a) shows period-1 control and (b) shows period-2 control.
The rotation rate for the
copper disk anode was 
2670 rpm, the anodic potential with
control off was 
0.724 V, and the values of the RPF proportionality
constants were K = -5.0 mV/mA and R = -0.3 for the period-1 control
shown in Fig. 2a. For the period-2 control shown in
Fig. 2b, 
2160 rpm, 
0.745 V, K = -5.0 mV/mA,
and R = 0.21. As described in reference[2],
we chose a region of
parameter space where the electrochemical
system exhibits a sequence of periodic mixed mode oscillations separated by
bands of chaotic behavior as a function of anodic potential.
The period-1 control was done while in the chaotic region between a period-1
state with large amplitude oscillations only and a period-2 state with one
large amplitude oscillation followed by one small oscillation.
We were unsuccessful in controlling on the period-2 oscillation in this
chaotic band. Successful control of a period-2 state was attained
in the chaotic band between the period-2 (one large--one small) and
period-3 (one large and two small) mixed mode oscillations. We
found that control of the period-2 orbit was much more difficult than the
period-1. This is partially because the period is about twice as long
(about 5 sec) and the feedback corrections are made just once each cycle.
Thus the system can drift further away from its fixed point before the
next feedback correction is calculated and applied.
The corresponding return maps for the two cases are shown in Fig. 3 while Fig. 4 and 5 show the chaotic attractors and controlled periodic orbits reconstructed from the time series of the anodic current using a two-dimensional time delay embedding.
Figure: (a) The first iterate return map (open circles) obtained
from the sequence of current minima
for the chaotic state shown in Fig. 2a. The superimposed filled circles
are the minima in the anodic current while the control algorithm was
implemented. (b) the corresponding
second-iterate return map for the period-2 case shown in Fig. 2b.
Figure: (a) The two-dimensional time delay embedding
with 
120 msec
showing the chaotic
attractor for the situation shown in Figs. 2a and 3a. (b) the corresponding
period-1 trajectory with control on.
Figure: (a) The two-dimensional time delay embedding
with 
120 msec
showing the chaotic
attractor for the situation shown in Figs. 2b and 3b. (b) the corresponding
period-2 trajectory with control on.